Fibring

Origin. Dov Gabbay, "Fibred semantics and the weaving of logics" (Journal of Symbolic Logic, 1996) and Fibring Logics (Oxford Logic Guides 38, 1999). Developed algebraically by Sernadas, Sernadas, and Caleiro (1999), whose categorical account made the collapsing problem visible. The most general of the combination methods: it joins arbitrary logics, not just modal ones.

Models. Two logics, two languages, one combined language in which each may occur inside the other's formulas. Semantically, a model of the combination attaches to each point of one logic's structure a whole model of the other — the fibre. The construction is a coproduct: it adds nothing beyond what the components had, which is exactly why it can collapse.

Formalism.

Unconstrained (free) fibring: Signatures Σ₁, Σ₂; the combined signature is Σ₁ ∪ Σ₂. Formulas mix freely: □(p →₂ q) is well formed when □ ∈ Σ₁, →₂ ∈ Σ₂. Consequence: the least consequence relation extending both, closed under both sets of rules.

Fibred semantics (Gabbay): A model of L₁ ⊕ L₂ assigns to each point w of an L₁-structure an L₂-model M(w) — the fibre at w. Evaluation switches structures at the connective boundary: an L₂-formula at w is evaluated in M(w). The fibring function is the glue.

Constrained fibring: Identify some symbols across the components — usually the classical base. Shared connectives are shared, not duplicated. Most useful combinations are constrained; unconstrained fibring of two classical-based logics duplicates ⊥, ¬, →.

Algebraic fibring (Sernadas et al.): Logics as objects in a category, fibring as the coproduct. The universal property is what makes it "the" combination and what makes the collapse inevitable.

Dovetailing: Gabbay's variant, distinguished by name from fibring proper — L^F_I versus L^D_I — and the source of the transfer results for combining arbitrary, possibly non-normal, modal or intermediate logics complete for a class of Kripke models: recursive axiomatizability, decidability, and the finite model property all transfer. Governatori, Nair, and Sattar (2002) use dovetailing to build BDI logics, constraining the fibring function to capture interaction axioms of Catach's schema. The fibring/dovetailing distinction is drawn differently across the literature, and the secondary sources describe dovetailing only as "a variant" or "a special form" of fibring without a stable definitional contrast. That is itself a fact about the subject: fibring is defined on presentations rather than on consequence relations, so its variants multiply and none of the boundaries between them is canonical.

The collapsing problem: Fibre classical logic with intuitionistic logic, sharing the connectives. The combination proves excluded middle for the intuitionistic connectives. The intuitionistic component is destroyed by contact. Diagnosis (Caleiro, Ramos): the shared connectives inherit both sets of rules, and the classical rules are stronger. Fixes: modulated fibring, cryptofibring, keeping the components' connectives distinct.

Transfer: Completeness transfers under conditions (Zanardo, Sernadas, Sernadas 2001). Decidability does not in general.

Symbols.

Metatheory. The collapsing problem is the subject's central result and its central embarrassment: the most natural combination of two logics can annihilate one of them, and the annihilation is a consequence of the universal property rather than a defect in the definition. That the coproduct is the right categorical answer and the wrong logical one is what generated the modulated and cryptofibring variants, none of which is canonical. Completeness transfer holds under a fullness condition on the component semantics; decidability transfer fails, which is the standard pattern in this area — the syntactic properties travel and the algorithmic ones do not.

Applies to. Combining modal with temporal, epistemic with deontic, classical with substructural. Modular specification, where a system's logic is assembled from components. The theoretical question of what a logic is, which fibring makes concrete by asking what survives when two are glued.

Limitations. Collapse makes unconstrained fibring useless for the cases most wanted — combining a classical logic with a weaker one is exactly where it fails. The repairs are ad hoc: each blocks the collapse by a different device, and no principled account says which to use. Fibring is defined for logics presented in a particular way, and two presentations of the same consequence relation can fibre differently, so the operation is on presentations rather than on logics.

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